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Groupprops β

Degree of irreducible representation may be greater than exponent

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Statement

We can have a finite group and an irreducible linear representation of the group over an algebraically closed field of characteristic zero, such that the degree of the irreducible representation is greater than the exponent of the group.

This is a non-constraint on the Degrees of irreducible representations (?) of a finite group.

Related facts

Proof

Example of big extraspecial groups

Consider an extraspecial group of order p^7 for any prime p. The exponent of this group is either p or p^2. On the other hand, it admits a faithful irreducible representation of order p^3.